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June  2014, 6(2): 261-278. doi: 10.3934/jgm.2014.6.261

Periodic orbits in the Kepler-Heisenberg problem

Corey Shanbrom 1,

1. 

Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA, United States

Received  November 2013 Revised  March 2014 Published  June 2014

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with $k$-fold rotational symmetry for any odd integer $k\geq 3$. Approximations are shown for $k=3$.
Keywords: Heisenberg group, Periodic orbits, fundamental solution to Laplacian., integrable system, Kepler problem, Carnot group.
Mathematics Subject Classification: 53C17, 37N05, 37J45, 70H12, 70H0.
Citation: Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261
References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings, 1978. Google Scholar

[2]

A. Albouy, Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 13 (2008), 525-542. arXiv:math-ph/0501026v2. doi: 10.1134/S156035470806004X.  Google Scholar

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.  Google Scholar

[4]

D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012), 373-416.  Google Scholar

[5]

G. Bliss, The problem of Lagrange in the calculus of variations, American J. Math., 52 (1930), 673-744. doi: 10.2307/2370714.  Google Scholar

[6]

O. Bolza, Calculus of Variations, $2^{nd}$ edition, Chelsea, 1960.  Google Scholar

[7]

R. W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Appl. Math., (eds. P.J. Hilton and G.S. Young), Springer-Verlag, (1982), 11-27.  Google Scholar

[8]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357.  Google Scholar

[9]

F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria, Journal of Nonlinear Science, 22 (2012), 247-266. doi: 10.1007/s00332-011-9116-z.  Google Scholar

[10]

G. Folland, A fundamental solution for a subelliptic operator, Bulletin of the AMS, 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4.  Google Scholar

[11]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 1963.  Google Scholar

[12]

W. B. Gordon, A minimizing property of Keplerian orbits, American J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.  Google Scholar

[13]

P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983.  Google Scholar

[14]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels, in Collected Works, II (1949), 1835-1838, (Russian), GITTL, Moscow. Google Scholar

[15]

R. Montgomery, A Tour of Subriemannian Geometries, AMS, 2002. Google Scholar

[16]

R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere,, to appear in Geometry, ().   Google Scholar

[17]

R. Palais, The principle of symmetric criticality, Commun. Math. Phys, 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[18]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962.  Google Scholar

[19]

C. Shanbrom, Two Problems in Sub-Riemannian Geometry, Ph.D thesis, UC Santa Cruz, 2013.  Google Scholar

[20]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, $2^{nd}$ edition, Chelsea, 1980. Google Scholar

show all references

References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings, 1978. Google Scholar

[2]

A. Albouy, Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 13 (2008), 525-542. arXiv:math-ph/0501026v2. doi: 10.1134/S156035470806004X.  Google Scholar

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.  Google Scholar

[4]

D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012), 373-416.  Google Scholar

[5]

G. Bliss, The problem of Lagrange in the calculus of variations, American J. Math., 52 (1930), 673-744. doi: 10.2307/2370714.  Google Scholar

[6]

O. Bolza, Calculus of Variations, $2^{nd}$ edition, Chelsea, 1960.  Google Scholar

[7]

R. W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Appl. Math., (eds. P.J. Hilton and G.S. Young), Springer-Verlag, (1982), 11-27.  Google Scholar

[8]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357.  Google Scholar

[9]

F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria, Journal of Nonlinear Science, 22 (2012), 247-266. doi: 10.1007/s00332-011-9116-z.  Google Scholar

[10]

G. Folland, A fundamental solution for a subelliptic operator, Bulletin of the AMS, 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4.  Google Scholar

[11]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 1963.  Google Scholar

[12]

W. B. Gordon, A minimizing property of Keplerian orbits, American J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.  Google Scholar

[13]

P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983.  Google Scholar

[14]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels, in Collected Works, II (1949), 1835-1838, (Russian), GITTL, Moscow. Google Scholar

[15]

R. Montgomery, A Tour of Subriemannian Geometries, AMS, 2002. Google Scholar

[16]

R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere,, to appear in Geometry, ().   Google Scholar

[17]

R. Palais, The principle of symmetric criticality, Commun. Math. Phys, 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[18]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962.  Google Scholar

[19]

C. Shanbrom, Two Problems in Sub-Riemannian Geometry, Ph.D thesis, UC Santa Cruz, 2013.  Google Scholar

[20]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, $2^{nd}$ edition, Chelsea, 1980. Google Scholar

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